Tegime EJSprogrammiga kokku neli erinevat parameetrite hindamist, parametrisatsioonidega CPL, MZ I, MZ II ja LZ. Hindamine k¨aib n˜onda, et k˜oigile uuritavatele parameetritele vali-takse mingi vahemik, milles neid hinnavali-takse ning samm, mille vahemikus liiguvali-takse. Teades parameetrite eeldatavaid v¨a¨artusi, valisin vahemikud j¨argnevalt:
Ωminm = 0.260, Ωmaxm = 0.320, wmin0 =−0.950, wmax0 =−1.050, wmin1 =−0.100, wmax1 = 0.100, Gmin1 =−0.005, Gmax1 = 0.005. Samm oli k˜oigil parameetritel 0.001.
Lisas (vt allpool) on antudχ2arvutamise t¨apne kood, kus on n¨aha, et k˜oigepealt arvutatakse minimaalsete parameetri v¨a¨artustega k˜oigi 580 supernoova jaoks teoreetiline kaugusmoodul, millede abil saab χ2 v¨a¨artuse. Seej¨arel t˜ostetakse G1 v¨a¨artust sammu v˜orra ning arvutatakse χ2 v¨a¨artus uuesti. N˜onda tegutsetakse, kuni G1 saavutab oma maksimaalse v¨a¨artuse, misj¨arel l¨aheb see miinimumi tagasi ning w1v¨a¨artust t˜ostetakse sammu v˜orra. Niiviisi k¨aiakse l¨abi k˜oik v˜oimalikud parameetrite kombinatsioonid ning leitakse v¨ahimχ2 v¨a¨artus.
Tabel 1.Parim tulemus iga parametrisatsiooni korral
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Parametrisatsioon
Parameeter
Ωm w0 w1 G˜1 χ2SN/ν
CPL 0.285 −1.011 −0.011 −0.002 0.9761
MZ I 0.285 −1.009 0.005 0.002 0,9761
MZ II 0.278 −1.025 0.044 −0.003 0.9761
LZ 0.285 −1.009 0.006 −0.003 0.9761
Tabelis t¨ahistabν vabadusastmete arvu, mis meie m˜o˜otmiste puhul on vabade parameetrite arv lahutatud m˜o˜odetavate supernoovade arvust (ν = 580−4 = 576). N¨aeme, etχ2/ν ≈ 1, nagu peakski tulema [41]. V˜ordluseks lisame artiklis [28] saadud minimaalseleχ2’le vastavate parameetrite v¨a¨artused. Nende tulemuste arvutamiseks kasutati s¨umbolarvutusprogrammi Ma-ple.
Tabel 2.Artiklis [28] saadud tulemused
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Parametrisatsioon
Parameeter
Ωm w0 w1 G˜1 χ2
SN/ν
CPL 0.280 −1.008 −0.016 0.001 0.9761
MZ I 0.279 −1.007 0.011 0.001 0,9761
MZ II 0.281 −1.012 0.015 0.001 0.9761
LZ 0.289 −0.970 0.158 0.003 0.9760
6 Arutelu
J¨argnevalt arutleme saadud hindamistulemuste ja t¨o¨o j¨atkamise v˜oimaluste ¨ule.
Tabelis 1 leitud v¨a¨artustest n¨aeme, et leitud parameetrid s˜oltusid m¨argatavalt valitud tume-energia parametriseeringust. On n¨aha, et kolme parametriseeringu korral onG1 < 0ehk efek-tiivne gravitatsioonikonstant on kahanev, mis toetab universumi kiirenevat paisumist. Lisaks n¨aeme, et k˜oigil juhtudel tumeenergia olekuv˜orrandi parameeterw0 <−1, mis vastab fantoom-energiale. Universumi energiatihedus kasvaks ning kosmiline l˜oppstsenaarium v˜oiks olla min-git t¨u¨upi rebend (vt t¨apsemalt peat¨ukk (3.3)). Tuleb aga arvesse v˜otta m˜o˜otmistulemuste suu-ri m¨a¨aramatusi (viimased kajastuksid kontuurjoonisel suurte piirkondadena parameetsuu-rite stan-dardh¨alvete jaoks) ning asjaolu, et leidsime vaid minimeeritudχ2 v¨a¨artuse.
N¨aeme, et k¨aesolevas t¨o¨os saadud tulemused on k¨ull l¨ahedased artiklis [28] saadud tu-lemustega, kuid siiski esinevad erinevused. Erinevused tulevad ilmselt sisse teoreetilise kau-gusmooduli erinevatest arvutusviisidest, kuid v˜oivad olla p˜ohjustatud ka erinevate programmi-de erip¨araprogrammi-dest (χ2 arvutamisel kasutatakse v¨ahemalt k¨umnendasse komakohta ulatuvaid arve).
Union 2.1andmete kohta Suzukiet alpoolt tehtud anal¨u¨usiga [40] on tulemusi raskem v˜orrelda, kuna seal ei kasutatud d¨unaamilist gravitatsioonikonstanti, ega ka olekuparameetri selliseid pa-rametriseeringuid. Fikseeritud w1 = 0korral olid Ωm = 0.281 ja w0 = −1.011 [40], mis on siiski ¨upris l¨ahedased ka k¨aesolevas t¨o¨os saadud tulemustega.
Kahjuks ei olnud t¨o¨o kirjutamise hetkel olemas uuemaid supernoovade andmeid kuiUnion 2.1 (publitseeritud 2011. aastal). Samas on hetkel k¨aimas m˜o˜otmised Euroopa Kosmoseagen-tuuriPlanck’i sateliidiga, mis m˜o˜odab kosmilist taustkiirgust. Nende andmete abil on v˜oimalik anda kosmoloogilistele parameetritele t¨apsemaid hinnanguid [15]. Lisaks oleks v˜oimalik kom-bineerida supernoovade, kosmilise taustkiirguse ning bar¨uonide v˜onkumise vaatlusandmeid, mis v˜oimaldaksid oluliselt kitsendada parameetrite varieeruvust.
7 Kokkuv˜ote
K¨aesolevas t¨o¨os tuletasime esmalt Einsteini v¨aljav˜orrandid, mille abil saime tuletada Friedman-ni v¨aljav˜orrandid. FriedmanFriedman-ni v˜orrandid m¨a¨arasid FriedmanFriedman-ni kosmoloogia, mida hiljem ka-sutasime kosmoloogiliste mudelite kirjeldamiseks. Lisaks uurisime Hubble’i parameetrit, mis kirjeldab ruumi paisumist ning mida on v˜oimalik leida t¨ahtedelt meieni j˜oudva valguse puna-nihke kaudu. T¨o¨o esimeses osas defineerisime ka kosmoloogilised parameetrid, nagu barotroop-ne indeks ja tihedusparameetrid. Need parameetrid m¨a¨aravad universumi d¨unaamika, olebarotroop-nevalt kosmoloogilisest mudelist.
T¨o¨o teises osas k¨asitlesime tumeainet ja tumeenergiat, mis eeldatavalt moodustavad ligikau-du 95% kogu universumi energiatiheligikau-dusest. Kuni tumeenergia on domineeriv energialiik, j¨atkab ΛCDM mudeli j¨argi universum kiirenevat paisumist. Kui juhtub, et energiatihedus ajas kasvab, siis v˜oivad tekkida kosmilised rebendid, mille klassifikatsioonidest oli antud alajaotuses juttu.
Kuigi standardmudelit kirjeldav ¨uldrelatiivsusteooria on tehtud vaatlustega heas koosk˜olas, on siiski ka standardmudelis lahendamata probleeme, millest m˜oned said kirjeldatud ka t¨o¨o esi-meses osas. Seet˜ottu otsitakse standardmudelile alternatiive, millest ¨uhte tutvustasime k¨aesoleva t¨o¨o kolmandas osas. Selleks on skalaar-tensor t¨u¨upi gravitatsiooniteooria.
T¨o¨o viimases osas kasutameUnion 2.1poolt m˜o˜odetud Ia-t¨u¨upi supernoovade vaatlusand-meid kosmoloogiliste parameetrite hindamiseks. Hinnatakse mateeria energiatihedust (mis on seotud tumeenergia tihedusega), tumeenergia olekuv˜orrandit ning efektiivset gravitatsiooni-konstanti. Parameetreid hinnatakse nelja erineva Hubble’i parameetri parametrisatsiooni kor-ral. Hindamine k¨aib programmiga Easy Java Simulations, mida modifitseerisime minimaalse χ2 leidmise jaoks, lisades ka vajalike parametriseeringute arvutamise meetodid. Arutlus saadud tulemuste ¨ule on toodud t¨o¨o l˜opus.
8 T¨anus˜onad
Tahan s¨udamest t¨anada k˜oiki, kes aitasid kaasa selle t¨o¨o valmimisele. T¨anan juhendajat Margus Saali, kes suurendas minu huvi relatiivsusteooria ja kosmoloogia vastu juba erirelatiivsusteooria kursustel ning kes aitas p¨uhendumisega kaasa selle t¨o¨o jaoks vajalike teadmiste omandamisele.
Lisaks t¨anan Andrjus Frantskjavitˇsiust, kes aitas kaasa hindamisprogrammi valmimisele.
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[41] http://maxwell.ucsc.edu/ drip/133/ch4.pdf ,Chi-square: Testing for Goodness Fit
Finding cosmological density parameters and the dynamic gravitational constant from type-Ia supernovae observations
In this paper, we first derived Einstein’s field equations, which helped us derive Friedmann’s field equations. Friedmann’s equations were used to describe different cosmological models through Friedmann cosmology. In addition we scrutinized the Hubble parameter, which gives us the normalized rate of expansion for the universe. In the first part of the paper, we also defined cosmological parameters, such as the barotropic index and density parameters. Depending on the cosmological model, these parameters define the dynamics of the universe.
In the second part, we took a look at dark energy and dark matter, which supposedly make up around 95% of the energy density of the universe. According to the ΛCDM model, the universe will keep expanding at an accelerated rate, as long as dark energy is the dominant form of energy. If energy density does increases, then cosmological rips (of which we also talked in the second section) are possible.
Even though the theory of general relativity, which describes the standard model, is in good accordance with observations, there are still unsolved problems within the standard model. So-me of these problems are also described in the first part of this paper. Due to these problems, alternative models are looked for, one of which we took a closer look at in the third section of this paper. An alternative theory is the scalar-tensor gravitational theory.
The final chapter describes, how we used tpye-Ia supernovae data, collected byUnion 2.1, to fit cosmological parameters. Parameters being fitted include the energy density of matter (which is dependent of dark energy’s density), the dark energy equation of state parameter and the effective gravitational constant. Four different parametrizations were used for parametrising the Hubble parameter.Easy Java Simulationsis the program used for fitting, which we modified by adding functions for finding the minimizedχ2 and for necessary parametrizations. Conclusion of results is at the end of the final section.
Lisa
Globaalsed muutujad double c = 2.998E5;
double zmin = 0.0;
double h = H_0/100.0;
double mu_0 = 0.0;
double Omega_v = 1.0 - Omega_m;
double hc = c/H_0;
long[] factorials = new long[10002];
long[] sums = new long[590];
//The following variables are all defined depending on the parametrization
double G1, G1_max, w0, w0_max, wa, wa_max, Omega_m, Omega_max;
Dimensioonitu Hubble’i parameetri arvutamine
public class InverseH implements Function {
// This is where we define the model of the Universe double E;
public double evaluate(double x) { /*
//CPL
double wz = 3.0*(1.0 + w0 + wa);
double we = (-3.0*wa*x)/(1.0 + x);
double qz = Math.pow((1.0 + x),wz)*Math.exp(we);
E = Math.sqrt(Omega_m*Math.pow(1.0 + x, 3) + Omega_v*qz);
//MZ I
double wz = 3.0*(1.0 + w0 + wa*(1 - Math.log(2.0)));
double we = (-3.0*wa*(x + 2.0)*Math.log(x + 2.0))/(x + 1.0);
double qz = Math.pow((1.0 + x), wz)*Math.exp(we);
E = Math.sqrt(Omega_m*Math.pow(1.0 + x, 3) + Omega_v*Math.pow(2, 6*wa)*qz);
//MZ II
double cs1 = 0.5772156649 + Math.log(1 + x) + sums[snumber];
double cs2 = 0.5772156649 + Math.log(1) + sums[snumber];
double wz = 3.0*(1.0 + w0 - wa*Math.sin(1.0));
double we = 3.0*wa*(cs1 - cs2 - (Math.sin(x + 1.0))/(x + 1.0) + Math.sin(1.0));
double qz = Math.pow((1.0 + x), wz)*Math.exp(we);
E = Math.sqrt(Omega_m*Math.pow(1.0 + x, 3) + Omega_v*qz);
*/
//LZ
double wz = 3.0*(1.0 + w0);
double we =
((2.0/Math.PI)*wa*(Math.cos((3*Math.PI/2)*Math.log(1.0 + x)) -1.0));
double qz = Math.pow((1.0 + x), wz)*Math.exp(we);
E = Math.sqrt(Omega_m*Math.pow(1.0 + x, 3) + Omega_v*qz);
return 1.0/E;
} }
Integraalse koosinuse summaliikme arvutamine public void factoring(int n, int k) {
long fct = 1;
factorials[0] = 1;
for (int i = 1; i <= n; i++) { fct = fct * i;
factorials[i] = fct;
}
for (snumber = 0; snumber < supernovaedataset.length; snumber++) { long sum = 0;
double x = supernovaedataset[snumber][0];
//Calculates the cosine integral series for (int s = 1; s < k; s++) {
double top = Math.pow(-Math.pow(x, 2), s);
double bottom = 2*s*factorials[2*s];
sum += top/bottom;
}
sums[snumber] = sum;
} }
Teoreetilise kaugusmooduli arvutamine
public double integrate (double mu_0) {
// This is the integration routine we call for calculating the distance modulus
Function f = new InverseH();
//Check each possible case of Curvature (Omega_k).
if(Omega_k == 0) // If curvature is zero
D_L = (1.0 + z) * Integral.romberg(f, zmin, z, 1000, tol);
else{ // Otherwise
double OmegaK_root = Math.sqrt(Math.abs(Omega_k));
if(Omega_k < 0) // If positive curvature
D_L = (1.0 + z) * (1.0 / OmegaK_root) * Math.sinh(OmegaK_root * Integral.romberg(f, zmin, z, 1000, tol));
else // If negative curvature
D_L = (1.0 + z) * (1.0 / OmegaK_root) * Math.sin(OmegaK_root * Integral.romberg(f, zmin, z, 1000, tol));
}
double G_eff = 1 - G1 * Math.sin((2 * Math.PI * Math.log(1.0 + x))/Math.log(2));
//Theoretical distance modulus
return 5.0*Math.log10(D_L) + mu_0 + (15/4)*Math.log10(G_eff);
}
χ2 arvutamine
public void chi_sn() { double A;
double B;
double C;
double w0_begin = w0;
double wa_begin = wa;
double omega_begin = Omega_m;
double G1_begin = G1;
double min_chi = 10000;
double best_w0 = 0;
double best_omega = 0;
double best_G1 = 0;
double best_wa = 0;
//Four loops, one for each parameter, that go through values from min to max,
//by the step given
for (double m = Omega_m; m <= Omega_max; m += 0.001) {
Omega_m = m;
for (double wi = w0; wi <= w0_max; wi += 0.001) { w0 = wi;
for (double wj = wa; wj <= wa_max; wj += 0.001) { wa = wj;
for (double G_temp = G1; G_temp <= G1_max; G_temp += 0.001) { G1 = G_temp;
diffs_sn = 0.0;
A = 0;
B = 0;
C = 0;
// Again, a for loop to use the data we read
for (int i = 0; i < supernovaedataset.length; i++) {
double xx = supernovaedataset[i][0]; //redshift
double mu_obs = supernovaedataset[i][1]; //distance modulus double error = supernovaedataset[i][2]; //error
// Now using the redshift data points in our integration z = xx;
// Call the integration double mu_th = integrate(0);
// Compare the difference between distance modulus of theory (muth)
// and the observational distance modulus double residual = mu_obs - mu_th;
A += Math.pow(residual, 2)/Math.pow(error, 2);
B += residual/Math.pow(error, 2);
C += 1/Math.pow(error, 2);
mu_0 = B/C;
}
diffs_sn = A - Math.pow(B, 2)/C;
if (diffs_sn < min_chi) { min_chi = diffs_sn;
best_w0 = w0;
best_omega = m;
best_G1 = G1;
best_wa = wa;
}
_println("chi sq: " + diffs_sn);
_println("omega_m: " + m + ", w0: " + w0 + ", wa: " + wa + ", G1: " + G1);
_println("---");
}
G1 = G1_begin;
}
wa = wa_begin;
}
w0 = w0_begin;
}
_println("best chi sq: " + min_chi);
_println("best omega_m: " + best_omega);
_println("best w0: " + best_w0);
_println("best G1: " + best_G1);
_println("best wa: " + best_wa);
Omega_m = omega_begin;
}